# Parametric Stability of Solutions to Systems of Linear Inequalities with Various Subsets of Parameters

In this paper, the problem of minimal matrix correction of a system of linear algebraic inequalities was consider with the use of minimax criteria. The initial problem (a system of linear inequalities without a given property)

$$Ax ≤ b, A ∈ \mathbb{R}^{m×n}, b ∈ \mathbb{R}^m$$

there is assigned the class of corrected problems describing the independent variation of initial data

$$(A + H)x ≤ b + h.$$

Among the corrected problem, we have to find a problem with the given property and "closest"to the initial problem and, in addition, with fixation (release of the correction) of various combinations of rows and columns of parameter matrix (A, b). As the criterion of proximity of the initial and corrected problems, we use the matrix norm $||H||_∞ = \underset{i,j}{max} |hij |$.

Determine the stability measure of a solution to consistent system of linear inequalities $x$ as the value of the least change of the parameters after which x is not a solution to the system. We have proved that the problem of finding a solution that is the most stable to changes of various submatrices in extended matrix of the coefficients of the linear system by minimax criterion is reduced to a linear programming problem.

We have proposed some method for constructing a hyperplane separating two finite sets of points in $\mathbb{R}^n$. There is an exactly defined subset of incorrected coordinates (fixed columns in the parameter matrix). For the constructed hyperplane the limit change of the parameters (corrected coordinates of the given points) that reserves the separation property has the greatest value.

Key words: matrix correction, stability of compatible system of linear inequalities, separating hyperplane, minimax criteria.